The proof is similar to the proof of Lemma 6 in [1]}. Note that \(*{W_j}_{q,p} \ge *{W_j}_{q,\infty } \ge 0\)  for \(p\ge 1\) . So we have
\(\frac{M}{\Gamma } \ge \frac{\prod _{j=1}^{d}\left\Vert W_{j}\right\Vert _{q,p}}{\prod _{j=1}^{d}\left\Vert W_{j}\right\Vert _{q,\infty }}=\prod _{j=1}^{d} \frac{\left\Vert W_{j}\right\Vert _{q,p}}{\left\Vert W_{j}\right\Vert _{q,\infty }} \ge \prod _{j=1}^{r} \frac{\left\Vert W_{j}\right\Vert _{q,p}}{\left\Vert W_{j}\right\Vert _{q,\infty }} \ge \left(\min _{j \in \lbrace 1, \ldots , r\rbrace } \frac{\left\Vert W_{j}\right\Vert _{q,p}}{\left\Vert W_{j}\right\Vert _{q,\infty }}\right)^{r},\) 
